Is a three-way fixed effects model equivalent to a triple difference estimator? I have a conceptual question about the fixed effects model and the difference-in-differences (DD) estimator. Since a two-way fixed effects model is equivalent to a DD estimator, I was wondering if running a three-way fixed effects model would be equivalent to the difference-in-difference-in-differences (DDD) estimator?
Suppose I have state, year, and age group variation. Would it work to just add state, year, and my age group as the three fixed effects using the reghdfe command in Stata and assume I am running a triple difference estimator (i.e., DDD model)? Also, is it an issue if my age group variable is a dummy?
Thanks so much!
 A: 
Since a two-way fixed effects model is equivalent to a DD estimator, I was wondering if running a three-way fixed effects model would be equivalent to the difference-in-difference-in-differences (DDD) estimator?

Technically, yes. But it isn't as simple as tossing in the fixed effects at each level and estimating a three-way interaction term. The second level interaction terms must remain.
Say you have three dimensions over which a treatment may vary: state $s$, year $t$, and age group $a$. Let's look at the difference-in-difference-in-differences estimator when the timing of the intervention is well-defined. Suppose state $A$ is exposed to a policy and state $B$ is not. In state $A$, the policy is intended to target individuals under the age of 35. The equation is as follows:
$$
y_{iast} = \beta_0 + \beta_1Treat_s + \beta_2Age_a + \beta_3Post_t + \beta_4 (Treat_s \times Post_t) + \beta_5 (Age_a \times Post_t) + \beta_6 (Treat_s \times Age_a) + \beta_7 (Treat_s \times Age_a \times Post_t) + u_{iast}
$$
where $Treat_s$ is equal to 1 for state $A$, 0 otherwise. $Age_a$ is equal to 1 if a person is under the age of 35, 0 otherwise. $Post_t$ is a standardized time indicator equal to 1 in all years after the policy goes into effect, 0 otherwise.
You may have seen the more general representation of this equation:
$$
y_{iast} = \gamma_{st} + \lambda_{at} + \eta_{as} + \delta D_{ast} + X_{iast}'\beta + u_{iast},
$$
which includes state-year effects, age-year effects, and age-state effects. Depending upon how you define the relevant variables, your model may drop some of the second level terms. In software, you could also include a concatenated version of state-year and/or age-year into the model. This may result in collinearity, though it shouldn't affect your estimate of $\delta$.
In practice, I would simply regress $y_{iast}$ on state-year interactions, age-year interactions, age-state interactions, and your policy dummy. The relevant state, age, and year fixed effects will be estimated for free. Note, $D_{ast}$ is your triple interaction term, just defined in a different way. Here, we instantiate the treatment dummy manually. Put differently, $D_{ast}$ is equal to 1 if it meets three conditions: (1) the state is in the treatment group, (2) the individual falls in the younger age category, and (3) it is a "post-treatment" time period. Manually coding this interaction is useful in settings where the "timing" of the policy isn't always well-defined.
In my opinion, I think it's a bit misleading to say this estimator parallels the two-way fixed effects estimator. The generalized difference-in-differences estimator regresses some outcome on unit fixed effects, time fixed effects, and a treatment dummy. Note: two-way fixed effects implies separate  unit and time effects—not a unit-time effect. In keeping with my previous example, a single state-year effect is not appropriate in a difference-in-differences setting. In fact, in the absence of individual data within states, a state-year effect would chew up all your degrees of freedom. In short, it's insufficient to claim that a three-way fixed effects equation is a difference-in-difference-in-differences estimator. In settings where we triple difference, we must attempt estimation of all lower-order interaction terms.

Also, is it an issue if my age group variable is a dummy?

No problem at all.
In the first equation, $Age_a$ is equal to 1 if the individual is below a certain age threshold, 0 otherwise.
